The Gila Cliff Dwellings are the great excursion from Silver City: a set of Mogollon rooms built into natural caves above a side canyon of the Gila River. The loop trail is short but steep enough to feel like a real arrival, and the setting is the reward as much as the stonework.
The drive from Silver City takes around two hours on winding mountain roads. Start early, check National Park Service conditions, and treat this as a full day rather than an add-on.